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Tremolo picking means picking a note fast and repeatedly to give the impression of a single, sustained note with a "trembling" feel to it. You hear this a lot in mandolin playing: since the mandolin has such poor sustain, players use tremelo picking to play melodic lines that require longer-sustaining notes.

It's not just playing really fast. A famous example is Eddie Van Halen's "Eruption". He's playing notes really fast throughout the whole song, but the tremolo-picking part is only from 0:30 to 0:37.

As a technique of altering the sound of a sustaining note, tremolo differs from vibrato in that tremolo modulates the volume of the note (rapidly sounding slightly louder, then slightly softer), while vibrato modulates the pitch of the note (rapidly sounding slightly shaper, then slightly flatter). Interestingly, Fender got these two backwards: the "tremolo bar" on Strats actually produces vibrato---at least, that was what it was intended to do; Leo never envisioned it as a "whammy" bar---while the "vibrato" effect on Fender amps actually simulates tremolo.